The Scot is a long-time admirer of the 24-year-old French international, who seems destined for a move from the DW Stadium after his attitude was questioned by Latics boss Roberto Martinez.
McLeish would not be drawn on the size of the offer, although Blues have already had an £8million offer rejected, and says he is now waiting on a response from their Premier League rivals.
He told a press conference on Wednesday: "We have made another bid for Charles and we wait to see if it's successful or not.
"We're awaiting Wigan's response. At the same time we're in the hands of the gods in terms of the other offers. If there are other clubs in for him, then we have to wait and see.
"But we've tried - and we've tried our best. The board have worked hard to make these kind of things happen."
Sunderland, managed by former Birmingham boss Steve Bruce, have also been linked with a move for the former Newcastle player, who was left out of the opening day 4-0 defeat against Blackpool after Martinez criticised his attitude.
Italian duo Juventus and Fiorentina are also reportedly interested.
However, the Wigan boss has dismissed reports of a rift between the pair and introduced N'Zogbia as a substitute in the closing stages of last night's 3-0 Carling Cup second round win at Hartlepool.
McLeish admitted N'Zogbia is just one of several targets - but is not considering any of them a done deal.
He added: "There are a few other players. Again, it's difficult getting them over the line. That's going to be the hardest part.
"The deadline comes on Tuesday at 5pm and I'm sure it will be a mad scramble again. Getting players in place is going to be the most difficult thing.
"I don't have to name my 25-man squad until after the window closes, so that will be dictated by players who arrive or don't arrive."
The Scot is expected to make several changes for Thursday's televised Carling Cup second round home match against Rochdale, giving the opportunity to several youngsters to make an impression.
One player who definitely will not figure is veteran striker Kevin Phillips, who could be out for another month with a thigh problem picked up before the start of the campaign.